#### by Ronald G. Douglas

#### I. Introduction

For me, the year 1978 was a good year! In the winter I delivered the Hermann Weyl Lectures [e3] at the Institute for Advanced Study in Princeton. In the summer I gave an invited talk at the International Congress of Mathematicians held in Helsinki. And in December, I met Paul Baum. I had to navigate the snowy roads between Stony Brook and Princeton, and I lost my luggage on the trip to Helsinki, but the year ended on a good note!

On the plane trip returning from Helsinki, my then-colleague
Jeff Cheeger
told me I should look up Paul Baum in connection with my work on __\( K \)__-homology.
Earlier that summer, for the same reason,
Michael Atiyah
had suggested
to Paul that he look me up. Both Paul and I had come up with concrete
realizations of __\( K \)__-homology, and Michael Atiyah believed relating the
realizations could be very important.

My work had started with a problem in abstract operator theory seeming
to have little contact with __\( K \)__-theory. It was the outcome of this joint
effort with
Larry Brown
and
Peter Fillmore,
now called BDF theory, about
which I spoke in Princeton and Helsinki. An extant
write-up can be found in
[e2],
[e3].

In the 60’s a topic of considerable interest among operator theorists concerned compact perturbations and properties of operators which held modulo them. In particular, a result of Herman Weyl and John von Neumann showed that self-adjoint operators with the same essential spectrum were unitarily equivalent modulo the compact operators. Here, essential spectrum is defined as limit points of the spectrum or eigenvalues of infinite multiplicity. People were interested in the analog of this result for normal operators. And, in particular, Paul Halmos had asked if one could characterize such operators. David Berg had shown that normal operators with the same essential spectrum were unitarily equivalent modulo compacts. But whereas an operator that is self-adjoint modulo the compacts must be a self-adjoint plus a compact, the analogous statement is no longer true for normal operators. The unilateral shift provides a counterexample. In particular, Paul Halmos raised two specific questions. First, when is an operator that is normal modulo the compacts the sum of a normal and a compact? Second, is the collection of sums of a normal operator and a compact operator closed in the norm topology? These are some of the problems that Larry Brown, Peter Fillmore, and I were investigating.

In tackling these questions, following earlier work of
Lewis Coburn,
we eventually focused on the associated short exact sequence of __\( C^* \)__-algebras
__\[
0\to\mathcal{K}(\mathcal{H})\to\mathcal{E}(T)\to C(X_T )\to0.
\]__
Here __\( \mathcal{K}(\mathcal{H}) \)__ is the two-sided ideal of compact
operators on __\( \mathcal{H} \)__, __\( T \)__ is an operator on the Hilbert space
__\( \mathcal{H} \)__ satisfying
__\[
[T,T^* ]=TT^* -T^* T\in\mathcal{K}(\mathcal{H}),
\]__
__\( \mathcal{E}(T) \)__ is the __\( C^* \)__-algebra generated by __\( T \)__ and __\( I \)__, and __\( X_T \)__is the essential spectrum of __\( T \)__ or the spectrum of __\( \pi(T) \)__ in the Calkin
algebra __\( \mathcal{L}(\mathcal{H})/\mathcal{K}(\mathcal{H}) \)__. We realized
that the collection of such extensions formed a commutative
semigroup with identity in a natural fashion and that the question of when
__\( T=N+K \)__ is equivalent to the question of when the corresponding extension
is trivial or splits. But of course, this only replaced one problem with
another problem; that is, when
is such an extension trivial?

One obstruction to triviality is the existence of a __\( \lambda \)__
not in the
essential spectrum of __\( T,\,\sigma_e (T) \)__, for which the Fredholm index of
__\( T-\lambda \)__ is not zero. (Recall that an operator is said to be Fredholm
if it has closed range and both its
kernel and cokernel are finite-dimensional. Moreover, its index is the
difference of the dimensions of the kernel and the cokernel.) My results
with Larry Brown and Peter Fillmore ultimately showed that the converse held;
that is, the extension is trivial if and only
if there is no such __\( \lambda \)__. But proving that took some doing.

In considering the analogous question of triviality for spaces __\( X\subseteq
\mathbb{C}^m \)__ or for __\( m \)__-tuples of essentially normal operators
__\( (T_1,\dots,T_m) \)__ that commute modulo the compacts, we realized that one
needed to consider tensoring the corresponding extension by
__\( M_m(\mathbb{C}) \)__. Eventually, this led us to the fact that the semigroup
was a group that had a natural pairing with __\( K \)__-theory. Ultimately,
this showed that such extensions yielded a concrete realization for odd
__\( K \)__-homology and that, for __\( X \)__ a subset of the plane, the
Fredholm index mentioned above was enough to decide whether or not an
extension was trivial.

Following
Alexander Grothendieck’s
introduction of algebraic __\( K \)__-theory,
Michael Atiyah and
Fritz Hirzebruch
defined topological __\( K \)__-theory, and
Michael Atiyah later showed that an abstraction of elliptic operators
yielded a concrete analytic realization of even __\( K \)__-homology. However, he
was unable to describe the equivalence relation. The BDF work provided a
realization of the odd __\( K \)__-homology group and a workable set of equivalence
relations. Using the relation between the odd and even __\( K \)__-homology groups,
one could complete Michael Atiyah’s
realization of the even __\( K \)__-homology. About the same time,
Gennadi Kasparov
was attempting to follow up on Michael Atiyah’s work to obtain an equivalence
relation and use it to study the Novikov conjecture, which was Atiyah’s
original motivation.

Paul Baum had started out trying to provide a concrete geometric realization
of __\( K \)__-homology for a space __\( X \)__. After several attempts, he had come up
with a picture that Michael Atiyah liked. It involved the notion of a
__\( \mathrm{spin}^c \)__-manifold, a notion known to be related to __\( K \)__-theory.
Let __\( X \)__ be a finite complex. For each group __\( K_i (X),\,i=0,1 \)__, one considers
every compact __\( \mathrm{spin}^c \)__ manifold __\( M \)__ with dimension of the same parity as
__\( i \)__, every complex vector bundle __\( E \)__ over __\( M \)__, and every continuous map __\( f \)__
from __\( M \)__ to __\( X \)__. The triples __\( (M,E,f) \)__ are the cycles for __\( K_i (X) \)__. Paul
Baum had an equivalence relation built from three steps. The critical
step was what Paul called “vector bundle modification”. It captures
Bott periodicity and the fact that __\( K_i (X) \)__ is periodic in __\( i \)__ with period 2.
One might consider Paul’s cycles to be analogous to those
in singular homology theory, with __\( (M,f) \)__ playing the role of singular
simplex and __\( E \)__ the role of multiplicity.

Michael Atiyah had shown the existence of __\( K \)__-homology using Spanier–Whitehead
duality and had provided a concrete realization for it. But to be useful
one needed to be able to recognize naturally occurring cycles for this
theory and know when they are equivalent. In BDF we had analytic cycles,
while in Paul Baum’s description we had
geometric ones. Michael Atiyah’s belief that an explicit isomorphism between
these two pictures could be very important led to our meeting.

#### II. The isomorphism of realizations

At the end of the
70’s, an airline flew from Providence, Rhode Island, to Islip,
New York, and back, stopping at New Haven and Bridgeport in Connecticut, before
returning to Providence. Paul and I flew those routes regularly, once
every month or two for the next year and a half. Initially most of our
time was devoted to explaining to each other our realizations of __\( K \)__-homology.
Since each of us had limited expertise in the other’s field, this exercise
required considerable effort, and the consumption of much wine and good food,
especially at a now-closed restaurant called Napoleon’s in Port Jefferson,
New York.

In Michael Atiyah’s realization of even __\( K \)__-homology, __\( K_0 (X) \)__, for __\( X \)__
a finite complex, he defined cycles in terms of an abstract notion of an
elliptic operator: __\( (T,\sigma_1 ,\sigma_2 ) \)__, where __\( \sigma_1 \)__ and __\( \sigma_2 \)__
are __\( * \)__-representations of __\( C(X) \)__ on Hilbert spaces
__\( \mathcal{H}_1 \)__ and __\( \mathcal{H}_2 \)__ and __\( T \)__ is a Fredholm operator from
__\( H_1 \)__ to __\( H_2 \)__ which intertwines __\( \sigma_1 \)__ and __\( \sigma_2 \)__ up to the
compacts. The covariance required of a homology theory arises from the
following construction. Let __\( f : X \rightarrow Y \)__ be a continuous map and let
__\( (T,\sigma_1 ,\sigma_2 ) \)__ be an abstract elliptic operator on __\( X \)__. Define __\( f^*
: C(Y) \rightarrow C(X) \)__ by __\( f^* (\xi ) = \xi \circ f \)__. Then __\( (T,\sigma_1
\circ f^*,\sigma_2 \circ f^* ) \)__ is an abstract elliptic operator on __\( Y \)__. If
__\( X \)__ is a compact manifold, if __\( T \)__ is given by an elliptic pseudodifferential
operator of order 0
between sections of vector bundles __\( E_1 \)__ and __\( E_2 \)__ over __\( X \)__, and if
continuous functions on __\( X \)__ act on sections of these vector bundles
by pointwise multiplication, then one obtains an “Atiyah cycle” from
this data, and Michael Atiyah showed there were enough of those cycles
to generate __\( K_0 (X) \)__. However, he was unable to describe explicitly the
necessary equivalence relation.

Paul and I realized that the same setup with __\( \mathcal{H}_1 = \mathcal{H}_2 \)__,
__\( \sigma_1 =\sigma_2 \)__, and __\( T \)__ a self-adjoint Fredholm operator enables
one to obtain an odd cycle following BDF. In particular, if __\( P \)__ is the
orthogonal projection onto the positive spectral space for __\( T \)__, then
compressing a multiplier __\( \phi \)__ in __\( C(X) \)__ by __\( P \)__ to obtain a “Toeplitz-like
operator”, __\( P\sigma(\phi)P \)__, one obtains a unital __\( * \)__-homomorphism from
__\( C(X) \)__ to the Calkin algebra of the positive spectral subspace, which is
another version of a BDF cycle. In particular, this showed that the odd
analog of the Atiyah–Singer index theorem could be viewed as the classical
index theorem for Toeplitz operators such as those defined on the unit
circle by
Israel Gohberg
and
Mark Krein.
This observation also enabled us
to define the sought-after isomorphism between the analytic and geometric
realizations of __\( K \)__-homology.

Recall that for a finite-dimensional Hilbert space __\( G \)__, one can define the
associated Clifford algebra __\( \mathrm{Cl}(G) \)__ on which __\( G \)__ acts by multiplication. This
action is not irreducible since __\( \mathrm{Cl}(G) \)__ splits into an orthogonal direct
sum of submodules. Let __\( M \)__ be a smooth closed manifold with a Riemannian
metric. The fibers of the cotangent bundle, __\( T^*M \)__, to __\( M \)__ can be used
to define a bundle of complex Clifford algebras __\( \mathrm{Cl}(T^*M) \)__ over __\( M \)__. The
bundle __\( T^*M \)__ acts on __\( \mathrm{Cl}(T^*M) \)__ but not irreducibly. The statement that __\( M \)__
is a __\( \mathrm{spin}^c \)__ manifold means that not only does one have this
pointwise action on sections of the cotangent bundle but there is a sub-bundle
of the Clifford bundle __\( \mathrm{Cl}(T^*M) \)__, called the spinor bundle, on which __\( T^*M \)__
acts irreducibly. This enables one to define the Dirac operator, __\( D_M \)__,
on smooth sections of this spinor bundle to obtain a first-order
elliptic differential operator. When __\( M \)__ is odd-dimensional, the Dirac
operator is self-adjoint and the construction described in the preceding
paragraph, applied to __\( D(1 + D^2)^{-1/2} \)__, yields an odd BDF cycle. Moreover,
if we tensor __\( D \)__ with the identity operator __\( I_E \)__ on a vector bundle __\( E \)__,
we can proceed as above to make from __\( D\otimes I_E \)__ a BDF cycle constructed
from the pair __\( (M,E) \)__. A choice of connection on __\( E \)__ is required, but the
resulting odd cycle defines a class in __\( K_1(M) \)__ that is independent of the
choice of connection. Call this class __\( [D\otimes I_E] \)__. For a given triple
__\( (M, E, f) \)__ with __\( \dim M \)__ odd, using the covariance defined by composing
the __\( C(M) \)__-representation with __\( f^* \)__, we can
map __\( [D\otimes I_E] \in K_1(M) \)__ to a class
__\[
f_*([D\otimes I_E]) \in K_1
(X)
\]__
that is defined by a BDF cycle associated with an abstract elliptic
operator on __\( X \)__. The map
__\[
(M,E,f)\mapsto f_* ([D \otimes I_E] )
\]__
yields
the desired isomorphism between geometric and analytic odd __\( K \)__-homology.

In the setting of the preceding paragraph, but with __\( M \)__ even-dimensional,
the bounded operator constructed from __\( D\otimes I_E \)__ decomposes to
__\begin{equation*}
\begin{pmatrix}
0 & T^* \\
T & 0
\end{pmatrix}
\end{equation*}__
relative to the decomposition of the spinor bundle into even and odd spinors.
Using __\( T \)__, we obtain an Atiyah cycle on __\( M \)__ and we can push its class into
__\( K_0 (X) \)__, as before, to obtain the
map
__\[
(M,E,f)\mapsto (T,\sigma_0 \circ
f^*, \sigma_1 \circ f^*),
\]__
where the __\( \sigma_i \)__’s are the representations of __\( C(M) \)__ on the sections of
__\( M \)__’s even and odd spinor bundles. This map defines the desired isomorphism
in the even case.

By the time Paul and I attended the AMS Summer Institute in Operator Algebras in Kingston, Ontario, in the summer of 1980, we had understood the isomorphism and how to establish it. It is closely related to the Atiyah–Singer index theorem. These results were described in [1].

The Kingston meeting was Paul’s debut as an “operator algebraist”. After that he was no longer able to be anonymous in that community. He drove from Providence, took the ferry to Long Island, and picked me up at Stony Brook on the way to Kingston. He was one of the “geometer-topologists” who got interested in what is now known as noncommutative geometry. Alain Connes was at the Kingston meeting also.

#### III. Relative __\( K \)__-homology

I had planned to visit Alain Connes in Paris for the fall semester. As it
turned out, I could make it for only a week and Paul joined me. During
that time, Paul and Alain started work on what has become known as the
Baum–Connes conjecture. Paul and I, on the other hand, started working on
understanding an analytic realization for the even relative __\( K \)__-homology.
Some motivation for this was provided by the proof of the Atiyah–Singer
index theorem given in the Palais notes
[e1].
We got together frequently over
the next two or three years, working
to understand this realization. We also worked to understand the relation
of our work to that of Gennadi Kasparov, who was attempting to complete
Michael Atiyah’s original effort to resolve the Novikov conjecture. In his
work, a two variable “__\( KK \)__-theory” appeared and he had incorporated the
__\( C^* \)__-extension framework.

In defining __\( K \)__-homology theory there are essentially four groups: the even
group __\( K_0 (X) \)__, the odd group __\( K_1 (X) \)__, the even relative group __\( K_0
(X,A) \)__, and the relative odd group __\( K_1 (X,A) \)__, where __\( X \)__ is a compact
metrizable space and __\( A \)__ is a closed
subset of __\( X \)__. While it is possible to define the relative groups abstractly
in terms of the absolute groups, we believed that identifying a concrete
realization of relative cycles would lead to applications and new connections.
Our first announcement of what we thought to be
true was given at the U.S.-Japan operator algebras meeting held in Kyoto in
the summer of 1983. For Paul and for me this was a first visit to Japan.
On the way to Kyoto I stopped at Sendai at the invitation of
Masamichi Takesaki
before continuing to Kyoto.
In Kyoto we stayed at a Holiday Inn and we recounted our adventures,
both mathematical and otherwise, during breakfast each morning. Paul was
particularly intrigued by the Japanese restaurants and the “geisha-like
waitresses”. There was also a cafe on
the roof at which “post-World War II Hawaiian songs” were the order of
the day. The conference enabled us to discuss our relative __\( K \)__-homology
program with other participants at the workshop.

There are two parts to the realization of even relative __\( K \)__-homology: one for
the cycles in __\( K_0 (X,A) \)__ and a second for the boundary map __\( \partial:K_0
(X,A)\to K_1 (A) \)__. To be useful the latter map needs to be very explicit.
The even relative cycle
is a variation of the absolute Atiyah cycle __\( (T,\sigma_1 ,\sigma_2 ) \)__ in
which __\( T \)__ is now assumed only to be semi-Fredholm, which means the kernel
of __\( T \)__ can be infinite-dimensional. Additionally, one assumes that for
__\( \phi \in C(X) \)__ the compression of a multiplier
__\( \sigma_1 (\phi) \)__ to the kernel of __\( T \)__ is compact if __\( \phi \)__ vanishes on __\( A \)__.
This allows one to define the __\( K_1 \)__ cycle that is the image of the boundary
map as follows. Take __\( \xi \)__ in __\( C(A) \)__, extend it arbitrarily to __\( X \)__ to
obtain __\( \phi \)__ in __\( C(X) \)__, and compress multiplication by
__\( \sigma_1 (\phi) \)__ to the kernel of __\( T \)__. The resulting operator is
well-defined up to compacts and, with __\( Q \)__ denoting projection onto the
kernel of __\( T \)__, the map __\( \xi \mapsto Q \sigma_1 (\phi)Q \)__ defines a unital
__\( * \)__-homomorphism from __\( C(A) \)__ to the Calkin algebra of the kernel of __\( T \)__;
i.e., it defines a BDF cycle representing a class in __\( K_1 (A) \)__. This cycle
represents the image of __\( (T,\sigma_1 ,\sigma_2 ) \)__’s class under the desired
boundary map __\( \partial:K_0 (X,A)\to K_1 (A) \)__.

Showing that this definition yields the relative even group along with the
boundary map was tackled by Paul and myself during the special year 1984–85
at MSRI in Berkeley. Part of the proof rested on matching up these groups
with corresponding __\( KK \)__-groups of
Gennadi Kasparov. The program at MSRI was organized by Alain Connes,
Masamichi Takesaki, and myself. Because Paul had recently moved from Brown
University to Penn State, he was unable to join the group for the whole year.
However, he was a frequent visitor for several
weeks at a time.

#### IV. Differential operators and the relative even group

Although our work provides a rather explicit description of relative
even cycles and the boundary map, the description is not terribly useful
in the form given above. The problem is to decide how one recognizes a
relative cycle and the image of the boundary map when the cycle arises
from a differential operator on a smooth manifold. To accomplish this, we
were joined by
Michael Taylor,
who at that time was my colleague at Stony
Brook and who is an expert on
PDEs. We focused on differential operators,
and a generalization of them, called pseudodifferential operators, defined
from smooth sections of a vector bundle __\( E_0 \)__ to smooth sections of a vector
bundle __\( E_1 \)__ on a smooth manifold __\( M \)__ with boundary __\( \partial M \)__.

Since the differential operator defines an unbounded operator on the space
of smooth sections, one has to proceed carefully. In particular, one must
consider extensions of this operator that are closed. We showed that some
of them define even relative cycles. Because such operators have more than
one possible
closed extension, the issue of uniqueness arises, but under broad hypotheses
it turns out that the class represented by the relative cycle is independent
of the choice of closed extension. One can also identify the odd boundary
cycle explicitly, now as an elliptic pseudodifferential operator
__\( D_\partial \)__ on __\( \partial M \)__. The correspondence
__\[[D]\to[D_\partial ]\]__
defines the boundary map from __\( K_0 (M,\partial M) \)__ to
__\( K_1 (\partial M) \)__.

The fact that different closed extensions of __\( D \)__ define the same element
of __\( K_0 (M,\partial M) \)__ implies that their corresponding images, under
the boundary map, in __\( K_1 (\partial M) \)__ are equal, which has
far-reaching
consequences. For example, if __\( D \)__ is the Dirac operator __\( D_M \)__ on
__\( M \)__ (where we assume __\( M \)__ has a __\( \mathrm{spin}^c \)__ structure) then the image of __\( [D_M
] \)__ is the class defined by the Dirac operator on __\( \partial M \)__ defined by the
__\( \mathrm{spin}^c \)__ structure on __\( \partial M \)__ induced by the __\( \mathrm{spin}^c \)__ structure on __\( M \)__.
If one takes the maximal closed extension of __\( D_M \)__, this
image under the boundary map is the odd cycle defined by the
Calderón
projection. This identification yields the Toeplitz index theorem of
Boutet de Monvel
and, in fact, allows one to generalize it. Many more
results involving index theory follow from
this setup.

This realization is particularly interesting in
the case __\( X \)__ is a strictly
pseudoconvex complex manifold and __\( D \)__ is the __\( \overline{\partial} \)__-operator.
In this case, one can identify
the kernel __\( \overline{\partial} \)__ as the finite
direct sum of subspaces of mixed forms
with the first summand being the Bergman space on __\( X \)__ and with other
summands, which involve higher-degree forms, being finite-dimensional.
Thus, the Bergman space on __\( X \)__ is identified up to a finite-dimensional
subspace with the kernel of the
relative even __\( K_0 (X, \partial X) \)__ cycle. If one uses the above recipe to
calculate the boundary map, one finds that __\( [\overline{\partial}_{\partial
X}]=[D_{\partial X}] \)__. This enables one to conclude that the cycles
defined by
the Bergman space on __\( X \)__ and the Dirac operator on __\( \partial X \)__ represent
the same class in __\( K_1 (\partial X) \)__. Since the Atiyah–Singer index theorem
or the isomorphism that we obtained earlier describes the odd cycle defined
by the Dirac operator, one is able to calculate the odd cycle defined by
the Bergman space. This is essentially the result of Boutet de Monvel
at this level of generality, with a proof slightly different from the one
mentioned above.

This work, which was published in [2], was essentially completed for the 1988 AMS Summer Institute held in New Hampshire and organized by William Arveson and myself. Many of the themes of the subject now called noncommutative geometry were discussed there as well as at the meeting in Bowdoin, Maine, that followed. The location of the meetings was chosen to take advantage of the summer weather, but mother nature thwarted us. It was hot and fans were in short supply. However, one consequence of these meetings being held in New England was the opportunity to meet Paul’s father, an artist. I had already met his mother, also an artist, on several occasions because she lived in New York City. This opportunity provided some insight into Paul’s personality.

#### V. The odd relative group and concluding comments

It is also of interest to consider the odd relative group. Here the
relative cycles would be pairs __\( (T,\sigma) \)__, where __\( T \)__ is a symmetric
semi-Fredholm operator on a Hilbert space __\( \mathcal{H} \)__ and __\( \sigma \)__ is a
of __\( * \)__‑representation__\( C(X) \)__ on __\( \mathcal{H} \)__.
Again, one assumes that for a multiplier __\( \sigma (\phi ) \)__ arising from a
__\( \phi \in C(X) \)__ that vanishes on __\( A \)__ the compression of __\( \sigma (\phi ) \)__
to the kernel of __\( T \)__ is compact. One needs also to define the boundary map
effectively and the question is somewhat delicate, involving the classical
theory of extensions of
symmetric operators. Subsequent work by Michael Taylor finessed these
difficulties by taking a product __\( X\times S^1 \)__ but at a cost. In particular,
the boundary map is less explicit, as is matching up the boundary map in
the case of differential operators. Although
we discussed how to complete this development directly, we never carried it
out. Again, there are possible applications that I related to Sturm–Liouville
theory in higher dimensions.

It had been ten years at this point since I met Paul Baum. Although our
research interests diverged subsequently, we remain good friends and often
reminisce about our roles in the development of __\( K \)__-homology.